proof of Gauss’ mean value theorem

We can parameterize the circle by letting $z=z_0+r{e}^{i\varphi }$.Then $dz=ir{e}^{i\varphi }d\varphi$. Using the Cauchy integral formula we can express $f(z_0)$ in the following way:

$f(z_0)$$=$${\displaystyle \frac{1}{2\pi i}}{\displaystyle {\oint }_C}{\displaystyle \frac{f(z)}{z-z_0}}𝑑z$

$=$${\displaystyle \frac{1}{2\pi i}}{\displaystyle {\int }_0^{2\pi }}{\displaystyle \frac{f(z_0+r{e}^{i\varphi })}{r{e}^{i\varphi }}}ir{e}^{i\varphi }𝑑\varphi$

$=$${\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_0^{2\pi }}f(z_0+r{e}^{i\varphi })𝑑\varphi .$